Multi-objective Matroid Optimization with Ordinal Weights

TL;DR

This paper extends matroid optimization to ordinal objectives with multiple categories, providing polynomial-time algorithms for various problem variants and demonstrating their efficiency through numerical tests.

Contribution

It introduces new problem variants for ordinal matroid optimization and develops a polynomial-time solution method based on matroid intersection problems.

Findings

Efficient polynomial-time algorithms for ordinal matroid optimization.

Numerical validation on spanning tree and partition matroid problems.

Demonstrated practical effectiveness of the proposed approach.

Abstract

Bi-objective optimization problems on matroids are in general intractable and their corresponding decision problems are in general NP-hard. However, if one of the objective functions is restricted to binary cost coefficients the problem becomes efficiently solvable by an exhaustive swap algorithm. Binary cost coefficients often represent two categories and are thus a special case of ordinal coefficients that are in general non-additive. In this paper we consider ordinal objective functions with more than two categories in the context of matroid optimization. We introduce several problem variants that can be distinguished w.r.t. their respective optimization goals, analyze their interrelations, and derive a polynomial time solution method that is based on the repeated solution of matroid intersection problems. Numerical tests on minimum spanning tree problems and on partition matroids confirm the efficiency of the approach.